Canonical Behavior-Preserving Transformations for Static Linear Circuits A Boundary Normal-Form Theorem on n Nodes Using Only Current Sources and VCCS

Static linear lumped circuits (conductances, independent sources, and linear dependent sources, with no storage) can be studied through their boundary behavior: the set of boundary voltage–current pairs consistent with internal circuit laws. Fixing a set of accessible boundary nodes B of size n, and assuming standard well-posedness conditions for modified nodal analysis (MNA), we show that the boundary current injection vector iB depends affinely on the boundary voltage vector vB on an admissible affine set: iB = Yeq vB + i0 for all vB ∈ VB . We then provide a canonical boundary normal form that realizes this law using only indepen- dent current sources and voltage-controlled current sources (VCCS) connected directly to the boundary nodes. The construction is deterministic and idempotent, and it yields a complete classification: two circuits are behaviorally equivalent on the same boundary if and only if their normal-form parameters agree (modulo boundary constraints). A worked example (including a dependent source), an explicit VCCS synthesis list, and an exact numerical spot-check are included.